1. **Problem statement:** Find the radius $r$ of each circle given the perimeter or area of minor or major arcs/sectors. 2. **Formulas and rules:** - Perimeter of an arc (minor or major) = length of arc + 2r (the two radii forming the sector) - Length of arc = $r \times \theta$ where $\theta$ is in radians - Area of sector = $\frac{1}{2} r^2 \theta$ where $\theta$ is in radians - Convert degrees to radians: $\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$ - Given $\pi = \frac{22}{7}$ --- ### (a) Given: Perimeter of minor arc AB = 11.4 cm, angle $\theta = 72^\circ$ 3. Convert angle to radians: $$\theta = 72 \times \frac{22}{7} \times \frac{1}{180} = \frac{72 \times 22}{7 \times 180} = \frac{1584}{1260} = 1.257 \text{ radians}$$ 4. Perimeter of minor arc = arc length + 2r: $$11.4 = r \times 1.257 + 2r = r(1.257 + 2) = r \times 3.257$$ 5. Solve for $r$: $$r = \frac{11.4}{3.257} \approx 3.5 \text{ cm}$$ --- ### (b) Given: Perimeter of major arc AB = 52 cm, angle $\theta = 120^\circ$ 6. Major arc angle = $360^\circ - 120^\circ = 240^\circ$ 7. Convert to radians: $$\theta = 240 \times \frac{22}{7} \times \frac{1}{180} = \frac{240 \times 22}{7 \times 180} = \frac{5280}{1260} = 4.19 \text{ radians}$$ 8. Perimeter of major arc = arc length + 2r: $$52 = r \times 4.19 + 2r = r(4.19 + 2) = r \times 6.19$$ 9. Solve for $r$: $$r = \frac{52}{6.19} \approx 8.4 \text{ cm}$$ --- ### (c) Given: Area of minor sector OAB = 57.75 cm², angle $\theta = 300^\circ$ 10. Convert angle to radians: $$\theta = 300 \times \frac{22}{7} \times \frac{1}{180} = \frac{300 \times 22}{7 \times 180} = \frac{6600}{1260} = 5.24 \text{ radians}$$ 11. Area of sector formula: $$57.75 = \frac{1}{2} r^2 \times 5.24$$ 12. Solve for $r^2$: $$r^2 = \frac{2 \times 57.75}{5.24} = \frac{115.5}{5.24} \approx 22.04$$ 13. Solve for $r$: $$r = \sqrt{22.04} \approx 4.7 \text{ cm}$$ --- ### (d) Given: Area of major sector OAB = 61.6 cm², angle $\theta = 225^\circ$ 14. Major sector angle = $360^\circ - 225^\circ = 135^\circ$ 15. Convert to radians: $$\theta = 135 \times \frac{22}{7} \times \frac{1}{180} = \frac{135 \times 22}{7 \times 180} = \frac{2970}{1260} = 2.36 \text{ radians}$$ 16. Area of sector formula: $$61.6 = \frac{1}{2} r^2 \times 2.36$$ 17. Solve for $r^2$: $$r^2 = \frac{2 \times 61.6}{2.36} = \frac{123.2}{2.36} \approx 52.2$$ 18. Solve for $r$: $$r = \sqrt{52.2} \approx 7.23 \text{ cm}$$ --- **Final answers:** - (a) $r \approx 3.5$ cm - (b) $r \approx 8.4$ cm - (c) $r \approx 4.7$ cm - (d) $r \approx 7.23$ cm